# Inverse not always Unique for Non-Associative Operation

## Theorem

Let $\left({S, \circ}\right)$ be an algebraic structure.

Let $\circ$ be a non-associative operation.

Then for any $x \in S$, it is possible for $x$ to have more than one inverse element.

## Proof

Consider the algebraic structure $\left({S, \circ}\right)$ consisting of:

The set $S = \left\{{a, b, e}\right\}$
The binary operation $\circ$

whose Cayley table is given as follows:

$\begin{array}{c|cccc} \circ & e & a & b \\ \hline e & e & a & b \\ a & a & e & e \\ b & b & e & e \\ \end{array}$

By inspection, we see that $e$ is the identity element of $\left({S, \circ}\right)$.

We also note that:

 $\displaystyle \left({a \circ a}\right) \circ b$ $=$ $\displaystyle e \circ b$ $\displaystyle$ $=$ $\displaystyle b$

 $\displaystyle a \circ \left({a \circ b}\right)$ $=$ $\displaystyle a \circ e$ $\displaystyle$ $=$ $\displaystyle a$

and so $\circ$ is not associative.

Note further that:

 $\displaystyle a \circ b$ $=$ $\displaystyle e$ $\displaystyle$ $=$ $\displaystyle b \circ a$

and also:

 $\displaystyle a \circ a$ $=$ $\displaystyle e$

So both $a$ and $b$ are inverses of $a$.

Hence the result.

$\blacksquare$